# [Symplectic Geometry] QSMS SUMMER SCHOOL 2023

by qsms posted Jul 17, 2023
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일정시작 2023-08-21 2023-08-23 #7164D0

QSMS SUMMER SCHOOL 2023

DATE: 8/21 Mon  ~  8/23 Wed (3 days, 2 Lectures/day) 총 6강

PLACE:  129-104(Seoul National University)

SPEAKER; 오정석 (Imperial College London)

TITLE:  Gromov-Witten invariants and mirror symmetry

ABSTRACT:  Mirror symmetry or its understanding seems to get better even at this moment. But on the other hand it makes it looks too diverse to follow other's progress. In this talk I would like to introduce one old fashioned understanding in an enumerative geometer's point of view, following the work of Bumsig Kim.

The simplest version of mirror symmetry could be a symmetry of Hodge numbers of a pair of Calabi-Yau 3-folds. It says dimensions of tangent spaces of one's K\"ahler moduli and the other's complex moduli are the same. This predicts these two moduli spaces are isomorphic in local neighbourhoods. Over these two neighbourhoods, two different D-modules are naturally defined on each. Then an advanced version of mirror symmetry could be stated with an isomorphism between the two D-modules. These two define differential equations on the spaces of sections. Then mirror symmetry gives a relationship between the solutions, which are known as J and I-functions, respectively. The coefficients are generating functions of genus 0 Gromov-Witten invariants and period integrals, respectively.

In the above story, the former is completely understood in terms of genus 0 Gromov-Witten theory. Hence it can be generalised beyond Calabi-Yau 3-folds and genus >0. The latter is hard for enumerative geometers to understand because it is not developed with moduli spaces. But interestingly I-function can be written as a generating function of genus 0 quasimap invariants (Givental and Ciocan-Fontanine--Kim) though it is not fully understood why. The relationship between J and I-functions can be understood as a wall-crossing phenomenon of moduli spaces (Givental, Ciocan-Fontanine--Kim and others). So it seems quasimap theory plays some role in mirror symmetry.

Now quasimap theory defines a cohomological field theory for gauged linear sigma model (Favero--Kim). In other words, there is a curve counting theory for certain LG models, which can hopefully be connected to other progress in mirror symmetry.